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Guest Column: Math and the City

By Steven Strogatz May 19, 2009 8:26 pmMay 19, 2009 8:26 pm

Thanks again to Leon Kreitzman for four fascinating articles about biological clocks in everything from peonies to people. My sabbatical is rapidly drawing to a close — but it isn’t over yet! My guest for the next three weeks is Steven Strogatz, a professor of applied mathematics at Cornell University and the author of “The Calculus of Friendship: What a Teacher and a Student Learned about Life While Corresponding about Math,” to be published in August.
Please welcome him.
— Olivia

As one of Olivia Judson’s biggest fans, I feel honored and a bit giddy to be filling in for her. But maybe I should confess up front that, unlike Olivia and the previous guest writers, I’m not a biologist, evolutionary or otherwise. In fact, I’m (gasp!) a mathematician.

One of the pleasures of looking at the world through mathematical eyes is that you can see certain patterns that would otherwise be hidden. This week’s column is about one such pattern. It’s a beautiful law of collective organization that links urban studies to zoology. It reveals Manhattan and a mouse to be variations on a single structural theme.

The mathematics of cities was launched in 1949 when George Zipf, a linguist working at Harvard, reported a striking regularity in the size distribution of cities. He noticed that if you tabulate the biggest cities in a given country and rank them according to their populations, the largest city is always about twice as big as the second largest, and three times as big as the third largest, and so on. In other words, the population of a city is, to a good approximation, inversely proportional to its rank. Why this should be true, no one knows.

Even more amazingly, Zipf’s law has apparently held for at least 100 years. Given the different social conditions from country to country, the different patterns of migration a century ago and many other variables that you’d think would make a difference, the generality of Zipf’s law is astonishing.

Keep in mind that this pattern emerged on its own. No city planner imposed it, and no citizens conspired to make it happen. Something is enforcing this invisible law, but we’re still in the dark about what that something might be.

Many inventive theorists working in disciplines ranging from economics to physics have taken a whack at explaining Zipf’s law, but no one has completely solved it. Paul Krugman, who has tackled the problem himself, wryly noted that “the usual complaint about economic theory is that our models are oversimplified — that they offer excessively neat views of complex, messy reality. [In the case of Zipf’s law] the reverse is true: we have complex, messy models, yet reality is startlingly neat and simple.”

After being stuck for a long time, the mathematics of cities has suddenly begun to take off again. Around 2006, scientists started discovering new mathematical laws about cities that are nearly as stunning as Zipf’s. But instead of focusing on the sizes of cities themselves, the new questions have to do with how city size affects other things we care about, like the amount of infrastructure needed to keep a city going.

For instance, if one city is 10 times as populous as another one, does it need 10 times as many gas stations? No. Bigger cities have more gas stations than smaller ones (of course), but not nearly in direct proportion to their size. The number of gas stations grows only in proportion to the 0.77 power of population. The crucial thing is that 0.77 is less than 1. This implies that the bigger a city is, the fewer gas stations it has per person. Put simply, bigger cities enjoy economies of scale. In this sense, bigger is greener.

The same pattern holds for other measures of infrastructure. Whether you measure miles of roadway or length of electrical cables, you find that all of these also decrease, per person, as city size increases. And all show an exponent between 0.7 and 0.9.

Now comes the spooky part. The same law is true for living things. That is, if you mentally replace cities by organisms and city size by body weight, the mathematical pattern remains the same.

For example, suppose you measure how many calories a mouse burns per day, compared to an elephant. Both are mammals, so at the cellular level you might expect they shouldn’t be too different. And indeed, when the cells of 10 different mammalian species were grown outside their host organisms, in a laboratory tissue culture, they all displayed the same metabolic rate. It was as if they didn’t know where they’d come from; they had no genetic memory of how big their donor was.

But now consider the elephant or the mouse as an intact animal, a functioning agglomeration of billions of cells. Then, on a pound for pound basis, the cells of an elephant consume far less energy than those of a mouse. The relevant law of metabolism, called Kleiber’s law, states that the metabolic needs of a mammal grow in proportion to its body weight raised to the 0.74 power.

This 0.74 power is uncannily close to the 0.77 observed for the law governing gas stations in cities. Coincidence? Maybe, but probably not. There are theoretical grounds to expect a power close to 3/4. Geoffrey West of the Santa Fe Institute and his colleagues Jim Brown and Brian Enquist have argued that a 3/4-power law is exactly what you’d expect if natural selection has evolved a transport system for conveying energy and nutrients as efficiently and rapidly as possible to all points of a three-dimensional body, using a fractal network built from a series of branching tubes — precisely the architecture seen in the circulatory system and the airways of the lung, and not too different from the roads and cables and pipes that keep a city alive.

These numerical coincidences seem to be telling us something profound. It appears that Aristotle’s metaphor of a city as a living thing is more than merely poetic. There may be deep laws of collective organization at work here, the same laws for aggregates of people and cells.

The numerology above would seem totally fortuitous if we hadn’t viewed cities and organisms through the lens of mathematics. By abstracting away nearly all the details involved in powering a mouse or a city, math exposes their underlying unity. In that way (and with apologies to Picasso), math is the lie that makes us realize the truth.

The amount of heat a sphere dissipates is proportional to its surface-to-volume ratio. This ratio, however, is inversely proportional to its size. The same applies to mammals. The smaller the animal, the greater its surface compared to its volume, resulting in a greater loss in body heat. Therefore, small mammals need more calories to maintain their body temperature than large ones.

However, in mammals the heat loss does not increase as rapidly with diminishing size as with spheres. Zipf’s rule applies, because size is not the only factor. In multi-celled organisms, the cells relate and their interactions accrue synergism. As Max Wertheimer so aptly recognized, the whole is more than the sum of the pieces. I have written about Max Wertheimer’s discovery here://brainmindinst.blogspot.com/2008/06/professor-max-wertheimers-synergy.html

I have often wondered about the exponent 2 in the denominator of the Body-Mass Index. Geometry suggests it should be 3. If two bodies are scaled three-dimensional copies of each other, and made of the same materials (hence the same density), the mass should increase as the cube of the linear dimension. A six-footer should weigh 1.728 times as much as a five-footer. But for the same BMI, the six-footer should weigh only 1.44 times as much.
Do the theories of West, Brown, and Enquist, as cited by Professor Strogatz, shed any light on this question? Was the definition of BMI pulled out of the air, or does it have a sound theoretical basis?
Of course, the skin area varies with the square of the linear dimension, which is related to heat radiation, and hence metabolic rates. But that has to do with how long it takes a creature to eat its own body weight, not with height/weight ratios.

This is an interesting line of reasoning, and I think that features do recur in different systems exhibiting “organized complexity”, — whether they be brains, ant hills, bee hives, or cities — a topic that has been explored by Doug Hofstadter, Steven Johnson, and Jane Jacobs.

I would point out, though, that Dr. Strogatz should be careful about how he defines “city population” in arguing that a 1/n rule holds for the distribution of city sizes. Does he refer to the populations within the core city limits, the metro areas, the population within commuting distance, or none of these? If metro areas are considered, then the 1/n rule does not hold for the U.S. There are a large number of metro areas in the 4-6 million range that clash with the predicted 1/n distribution of sizes.

On an unrelated note, it would be lovely if governments could make residents in different municipalities bear the true costs of their infrastructure, including its environmental toll. It is true that dense cities make far more efficient use of their infrastructure and have lower rates of per capita greenhouse gas emissions. But because federal laws and local zoning codes have favored suburban, auto-centric development for the last 60+ years, most cities today actually end up saddled with higher costs of living than the hugely wasteful suburban areas which have proliferated across the country like weeds. Talk about the saints subsidizing the sinners.

I’m a little puzzled: I see the logic of the 3/4 power for 3D transport, but a city is basically 2D.

Nor does Zipf’s Law seem to hold for the US, at least when you use metro area population rather than artificial city boundaries. For instance, a quick search finds the New York metro area has about 19 million people, the LA Basin (Los Angeles out to San Bernadino. which is all one continuous urban area) is about 17 million. Then after a few inbetweens, there are a bunch of metro areas in the 4-5 million range…

Zipf’s Law is a classic example of the ability of humans to see patterns disirregardless.

One starts by selecting appropriate administrative districts (since the world’s countries are not all well defined), and then uses, from various population estimates, the ones that fit the model for cities inside those districts.

For example, what are the populations of New York City and Los Angeles? People who have their residences inside the five boroughs and inside the official LA city limits? People in a vaguely defined metropolitan area? One takes the population definition that fits the 2:1 ratio. ‘If it don’t fit, don’t use it.’ And Chicago is now 1/3 of NYC? What about when Chicago and LA were almost identical in size? How did one get 3:2:1?

I would have preferred to have seen this article come out at the beginning of last month, an excellent date for publishing these kinds of little known mathematical ‘facts.’

I’m just wondering about the dimensionality. I’d heard of the 3/4 scaling in living organisms before, though I haven’t seen the details of the argument. You mention it applies to three dimensional bodies, but then I’m surprised that it works for cities, which are more or less two dimensional.

Being a resident of a far northern city, Stockholm Sweden, I would bet that the city size rule is much stronger in cold climate countries than tropical countries. For mammals the cells require a set temperature and increased volumes hold the heat in much better and cheaper. When you grow the cells outside of the body the researcher supplies for free the warm set temperature. Up north the ability to sell and buy when the roads are snowed in, the wind is blowing and the availability of public transport in winter probably all are greatly aided by increased city size.

Zipf’s law isn’t quite that general. Three easily found exceptions in developed countries: Great Britain and France, where the biggest cities are far bigger than the second biggest, and Australia, where the biggest city is just bigger than the second. Exceptions in developing countries: Egypt, Nigeria and Brazil.
In other countries I looked at, such as the US, Japan and Spain the rule sort of holds. Should we call it Zipf’s rule of thumb?

How do you meaningfully define the boundaries of a “city”? This seems like something that you can force to be true by the way you define things. You could probably make a case that cities fall into any number of other distributions, all of which are mutually contradictory.

When mathematicians step outside the abstract, they are no more exact scientists than any social scientist.

If the first, second and third cities are half of the one above them, it seems to suggest that the cities are linked in some way for this to work. Its not population directly because cities don’t have a capacity limit. Instead what if it is economic capacity? Trade is influenced and influences all large cities, and is to some extent closed within national boundaries. I am guessing the 10, 5, 2.5 thing should be reflected in the way our industries and corporations manage their money and assets.

All of the above may be true in relation to gas stations and living organisms. But I live in Australia and the two largest cities – Sydney and Melbourne – are very close in size. Sydney has about 4.5 million and Melbourne just under 4 million people. The difference is less than 20%, not 100%.

Zipf’s law doesn’t hold in the three Baltic countries — Lithuania, Latvia and Estonia. Perhaps it’s because they haven’t been independant states for most of their recent history? The capital of Estonia — Tallinn has roughtly 400 000 inhabitants, the second largest city — Tartu, just about 100 000… I wonder if and how long the former colonies and provinces become independant preserve previous demographics.

Causality is a matter that is often very easy to show in academic circles, the crucial point is to show cause and effect which is much harder.
The examples offered are interesting, but gas stations, equals not all service services a or other activities, furthermore there is a difference between developed and developing world.
More interesting is to show path dependency, or the process involved, instead of a comparison on 2 independent variables which have nothing in common.
Depending on the choice one makes, it looks there is causality but is mere coincidence.

I’m sure I’m not the first person to notice this, but just for the record here are the population statistics (in millions) of the two largest urban areas of four countries I have visited recently. Readers will note that except for the fact that the largest city does indeed have a larger population than the second largest (which is true by definition), these figures hold absolutely no relation to “Zipf’s law.”

In both cases, Resource use ‘R’ scales with size S as R~S^0.75, roughly. R,S are metabolic rate and weight for mammals, and number of gas stations and population for cities.

But I see one potential problem – the mammalian law is the so-called Kleiber’s 3/4 law and (disutedly) comes from the necessity of distributing blood through a network in 3 dimensions – In West, Brown, and Enquist’s paper “space filling fractal networks of branching tubes.”

But a city is largely a 2 dimensional creature! Bigger cities are not deeper. Why should the exponent be the same in 2 and 3 dimensions? Maybe it is all just a coincidence.

Since then, the disparity has become even greater as Seoul has grown relative to other cities. Or, I am misunderstanding Zipf’s Law. Love to hear some feedback from any reader’s who understand it better than I.

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Olivia Judson, an evolutionary biologist, writes every Wednesday about the influence of science and biology on modern life. She is the author of “Dr. Tatiana’s Sex Advice to All Creation: The Definitive Guide to the Evolutionary Biology of Sex.” Ms. Judson has been a reporter for The Economist and has written for a number of other publications, including Nature, The Financial Times, The Atlantic and Natural History. She is a research fellow in biology at Imperial College London.